On mathematical theory of selection: continuous time population dynamics. (English) Zbl 1311.92137
Summary: Mathematical theory of selection is developed within the frameworks of general models of inhomogeneous populations with continuous time. Methods that allow us to study the distribution dynamics under natural selection and to construct explicit solutions of the models are developed. All statistical characteristics of interest, such as the mean values of the fitness or any trait can be computed effectively, and the results depend in a crucial way on the initial distribution. The developed theory provides an effective method for solving selection systems; it reduces the initial complex model to a special system of ordinary differential equations (the escort system). Applications of the method to the Price equations are given; the solutions of some particular inhomogeneous Malthusian, Ricker and logistic-like models used but not solved in the literature are derived in explicit form.
Keywords:
selection system; dynamics of distribution; the Price equation; inhomogeneous logistic model; replicator equationReferences:
| [1] | Ackleh AS, Marshall DF, Heatherly HE, Fitzpatrick BG (1999) Survival of the fittest in a generalized logistic model. Math Models Methods Appl Sci 9: 1379–1391 · Zbl 1008.92027 · doi:10.1142/S0218202599000610 |
| [2] | Ackleh AS, Fitzpatrick BG, Thieme HR (2005) Rate distribution and survival of the fittest: a formulation on the space of measures. Discret Contin Dyn Syst Ser B 5(4): 917–928 · Zbl 1078.92055 · doi:10.3934/dcdsb.2005.5.917 |
| [3] | Barton MN, Turelli M (1987) Adaptive landscapes, genetic distance and the evolution of quantitative characters. Genet Res 49: 157–173 · doi:10.1017/S0016672300026951 |
| [4] | Fisher RA (1999) The genetical theory of natural selection: a complete variorum edition. Oxford University Press, Oxford · Zbl 1033.92022 |
| [5] | Frank SA (1997) The Price equation, Fishers fundamental theorem, kin selection and causal analysis. Evolution 1712: 51 |
| [6] | Gause GF (1934) The struggle for existence. Williams and Wilkins, Baltimore. http://www.ggause.com/Contgau.htm |
| [7] | Gorban AN (1984) Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis, Nauka, Novosibirsk [in Russian] |
| [8] | Gorban AN (2007) Selection theorem for systems with inheritance. Math Model Nat Phenom 2(4): 1–45 · Zbl 1337.92150 · doi:10.1051/mmnp:2008024 |
| [9] | Gorban AN, Khlebopros RG (1988) Demon of Darwin: Idea of optimality and natural selection, Nauka (FizMatGiz), Moscow, [in Russian] |
| [10] | Grafen A (2006) A theory of Fisher’s reproductive value. J Math Biol 53: 15–60 · Zbl 1101.92031 · doi:10.1007/s00285-006-0376-4 |
| [11] | Goodnight KF (1992) The effect of stochastic variation on kin selection in a budding-viscous population. Am Nat 140: 1028–1040 · doi:10.1086/285454 |
| [12] | Haldane JBS (1990) The causes of evolution, Princeton Science Library. University Press, Princeton |
| [13] | Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, London · Zbl 0914.90287 |
| [14] | Karev GP (2000) The effects of non-uniformity in population models. Doklady Math 62: 141–144 |
| [15] | Karev GP (2003) Inhomogeneous models of tree stand self-thinning. Ecol Model 160: 23–37 · doi:10.1016/S0304-3800(02)00287-9 |
| [16] | Karev GP (2005a) Dynamic theory of non-uniform population and global demography models. J Biol Syst 13: 83–104 · Zbl 1124.91360 · doi:10.1142/S0218339005001410 |
| [17] | Karev GP (2005b) Dynamics of heterogeneous populations and communities and evolution of distributions. Discret. Continuous Dyn Syst Suppl Vol 487–496 · Zbl 1142.92034 |
| [18] | Karev GP (2008) Inhomogeneous maps and mathematical theory of selection. JDEA 14: 31–58 · Zbl 1136.92031 |
| [19] | Karev G, Novozhilov A, Koonin E (2006) Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics. Biology Direct 3 Oct 2006, 1:30 |
| [20] | Kotz S, Balakrishnan N, Jonson N (2000) Continuous multivariate distributions, vol 1, no 2. Wiley, New York |
| [21] | Lande R, Arnold SJ (1983) The measurement of selection on correlated characters. Evolution 37: 1210–1226 · doi:10.2307/2408842 |
| [22] | Lewontin RC (1974) The genetic basis of evolutionary change. Columbia University Press, NY |
| [23] | Li CC (1967) Fundamental theorem of natural selection. Nat Lond 214: 504 · doi:10.1038/214504a0 |
| [24] | Liebig J (1876) Chemistry applications to farming and physiology |
| [25] | Nikolaou M, Tam VH (2005) A new modeling approach to the effect of antimicrobial agents on heterogeneous microbial populations. J Math Biol 52: 154–182 · Zbl 1091.92053 · doi:10.1007/s00285-005-0350-6 |
| [26] | Novozhilov AS On the spread of epidemics in a closed heterogeneous population. Math Biosci (2009). doi: 10.1016/j.mbs.2008.07.010 |
| [27] | Page KM, Nowak MA (2002) Unifying evolutionary dynamics. J Theor Biol 219: 93 · doi:10.1016/S0022-5193(02)93112-7 |
| [28] | Perthame B (2007) Transport equations in biology. Birkhäuser, Basel · Zbl 1185.92006 |
| [29] | Poletaev IA (1966) On mathematical models of elementary processes in boigeocoenosis, problems of Cibernetics, No 16, pp 171–190 [in Russian] |
| [30] | Price GR (1955) The nature of selection. J Theor Biol 175: 389 · doi:10.1006/jtbi.1995.0149 |
| [31] | Price GR (1970) Selection and covariance. Nature 227: 520 · doi:10.1038/227520a0 |
| [32] | Price GR (1972) Extension of covariance selection mathematics. Ann Hum Genet 35: 484 · Zbl 0231.92006 · doi:10.1111/j.1469-1809.1957.tb01874.x |
| [33] | Rice SH (2006) Evolutionary theory. Sinauer Associates Inc., Sunderland, MA |
| [34] | Robertson A (1968) The spectrum of generic variation. In: Lewontin RC (ed) Population biology and evolution. University Press, Syracuse, p 4 |
| [35] | Schuster P, Sigmund K (1983) Replicator dynamics. J Theor Biol 100: 533–538 · doi:10.1016/0022-5193(83)90445-9 |
| [36] | Semevsky FN, Semenov SM (1982) Mathematical modeling of ecological processes, Gidrometeoizdat, Leningrad [in Russian] |
| [37] | Thieme HR (2003) Mathematics in population biology. Princeton University Press, Prinston · Zbl 1054.92042 |
| [38] | Wilson DS, Dugatkin LA (1997) Group selection and assortative interactions. Am Nat 149: 336–351 · doi:10.1086/285993 |
| [39] | Zeldovich KB, Chen P, Shakhnovich BE, Shakhnovich EI (2007) A first-principles model of early evolution: emergence of gene families, species, and preferred protein folds. PLoS Comput Biol 3(7) |
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